Semantics & Pragmatics Volume 10, Article 5: 1–24, 2017https://doi.org/10.3765/sp.10.5

Deriving the positive polarity behavior of plain disjunction*

Andreea C. NicolaeLeibniz-Zentrum Allgemeine

Sprachwissenschaft, Germany

Submitted 2016-03-24 / Decision 2016-05-25 / Revision received 2016-10-10 / De-

cision 2016-10-21 / Revision received 2016-10-27 / Accepted 2016-10-28 / Revision

received 2016-10-28 / Early access 2017-03-30 / Published 2019-10-9

Abstract The goal of this paper is to show that the positive polarity behavior

of plain disjunctions (e.g., French ou ‘or’) can be analyzed as an interplay

between a semantic requirement for obligatory exhaustification and an econ-

omy condition which prevents vacuous exhaustification, building on the

analysis provided by Spector (2014) to account for the PPI behavior of com-

plex disjunctions (e.g., French soit soit ‘either or’). I will argue that plain, but

not complex, disjunction allows the pruning of its conjunctive alternative,

using as evidence the contrast between these two types of disjunctions when

it comes to the optionality of their scalar implicature. I will show that once we

assume that exhaustification can take scope over a covert doxastic operator,

we can straightforwardly derive the unacceptability of plain disjunction PPIs

under negation, even in the absence of a scalar implicature.

Keywords: positive polarity, disjunction, exhaustification, scalar implicatures, covert

doxastic operator, uncertainty inferences

* I would like to thank Gennaro Chierchia, Luka Crnic, Anamaria Falaus, Danny Fox, AndreasHaida, Uli Sauerland and Benjamin Spector for the extensive discussions and encouragementreceived while working on this paper. This work has been presented and greatly benefitedfrom feedback received at Concordia, MIT and Utrecht, as well as at workshops held inJerusalem and Göttingen. I am also very grateful for the criticisms and suggestions madeby Kjell Johan Sæbø and two S&P anonymous reviewers. This research was funded by theGerman Federal Ministry of Research (BMBF Grant Nr. 01UG1411) and the German ResearchFoundation (DFG Grant SA 925/11-1) within the priority program SPP 1727 XPrag.de. All errorsare my own.

©2017 NicolaeThis is an open-access article distributed under the terms of a Creative Commons AttributionLicense (https://creativecommons.org/licenses/by/3.0/).

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1 Introduction

The analysis of positive polarity items (PPIs) has been the subject of muchdebate in recent literature, particularly from the point of view of how it canbe unified with the phenomena of implicature calculation more generally.This paper will contribute to this debate by offering an analysis of plaindisjunction PPIs (e.g., ou ‘or’ in French), especially as it relates to the polaritystatus of its kin, complex disjunction (e.g., soit soit ‘either or’ in French).

It has long been noted that plain and complex disjunction differ in termsof the strength of their scalar inference, with the inference ‘not both’ be-ing stronger for complex disjunction, either or, than for plain disjunction,or. Most recently, Spector (2014) has argued that this difference can be at-tributed to whether or not the scalar alternative of disjunction, conjunction,is obligatorily integrated into meaning, namely, whether disjunction triggersobligatorily exhaustification via the covert alternative-sensitive operator Exh(cf. Chierchia, Fox & Spector 2012). Complex disjunctions are also known toexhibit positive polarity behavior cross-linguistically, which Spector arguescomes for free once we invoke a notion of economy that takes exhaustifica-tion to be licensed only if it leads to a strengthened meaning. In this paperI show that his proposal, as it stands, cannot account for the distributionof plain disjunction PPIs, such as French ou, which I do in Section 4 afterintroducing the relevant data and background assumptions in Sections 2and 3. In Section 5 I show that once we supplement his proposal with theassumption that alternatives can be pruned when computing exhaustification(cf. Fox & Katzir 2011) as well as the claim that uncertainty implicatures canbe derived in the grammar (cf. Meyer 2013), the inability of plain disjunctionPPIs to receive narrow-scope readings with respect to negation will fall outstraightforwardly. This section will also discuss the curious ability of ou totake scope under certain DE operators as well as the fact that it can receivenarrow scope in the presence of two DE operators. Lastly, Section 6 concludesand discusses open questions.

2 Data of interest

It has been noted that disjunction exhibits polarity sensitivity in some lan-guages but not in others. Specifically, in certain languages disjunction canonly receive a wide scope interpretation with respect to negation. The dis-cussion that follows focuses exclusively on English and French, as these two

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Deriving the positive polarity behavior of plain disjunction

languages provide a clear contrast with respect to the available interpreta-tions that arise when disjunction occurs in the presence of negation. Startingwith English as the base case, we see that the plain disjunction or can receiveboth a narrow scope and a wide scope interpretation with respect to negation,given that (1) is ambiguous between (1a) and (1b).

(1) Mary didn’t invite Lucy or John for dinner.

a. Mary didn’t invite Lucy or she didn’t invite John for dinner.b. Neither Lucy nor John were invited to dinner by Mary.

The French plain disjunction ou, on the other hand, does not illustrate thesame ambiguity, with the reading in (2b), where the disjunction has narrowscope, being strongly dispreferred.1

(2) Marie n’a pas invité Léa ou Jean à dîner.‘Marie has not invited Léa or Jean for dinner.’

a. Mary didn’t invite Lucy or she didn’t invite John for dinner.b. ??Neither Lucy nor John were invited to dinner by Mary.

This inability of disjunction in certain languages to take narrow scope withrespect to negation has been dubbed the “anti-licensing” condition, and it con-stitutes one of three common diagnostics for predicting whether an item is aPPI. As discussed in Szabolcsi 2004, besides the property of “anti-licensing”by negation, another diagnosic for PPI-hood is the ability to be “rescued”: ifthe negation is itself in the scope of a downward entailing operator, then thePPI is claimed to be rescued, i.e., it can receive a narrow scope interpretationwith respect to the immediately c-commanding negation.2

(3) a. Tout étudiant qui n’a pas pris de cours de maths ou de physiquea raté l’examen.‘Every student who neither passed maths nor physics failed theexam.’

b. Si Paul n’avait pas invité Pierre ou Julie à dîner, cela aurait étéimpoli.

1 Some of the French data is from Spector 2014, and some obtained via personal communica-tion from Isabelle Charnavel, Alexandre Cremers and Jérémy Zehr.

2 There is considerable variation cross-linguistically with respect to which operators can rescuea PPI (see, e.g., Nicolae 2012) but for the purposes of this paper we will deal exclusively withthose in (3).

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‘If Paul had invited neither Pierre nor Julie for dinner, it would’vebeen rude.’

The goal of this paper is to explain the distributional properties of PPIdisjunctions using tools already employed elsewhere in the grammar. Thenext section offers a very brief overview of the grammatical approach toscalar implicatures, a theory of implicatures that has already been adopted toaccount for the behavior of other polarity-sensitive elements: NPIs (Chierchia2004, 2013), epistemic indefinites (Falaus 2010) and free-choice items (Fox2007).

3 The grammatical approach to scalar implicatures

I adopt the view that implicatures are derived in the grammar via a mechanismof exhaustification. The idea is that scalar elements activate alternatives andthe grammar integrates these alternatives in a systematic way within themeaning of the utterance. Chierchia, Fox & Spector (2012) (building on workin Krifka 1995, Chierchia 2004, Spector 2006, Fox 2007, among others) arguethat scalar implicatures are the result of a syntactic ambiguity resolutionin favor of an LF which contains a covert exhaustivity operator Exh, akin toonly. Exh negates only innocently excludable alternatives, namely alternativepropositions which can be negated without resulting in a meaning that entailsanother alternative proposition.

(4) Exh(p) = p ∧∀q ∈ IE(p,Alt(p)) : ¬q.where: IE(p,Alt(p)) = λq ∈Alt(p).¬∃r ∈Alt(p) : (p ∧¬q)→ r

Following Zimmermann 2000, Sauerland 2004, Alonso-Ovalle 2006, Spector2006 and Fox 2007, we assume that a disjunctive proposition such as (5) isassociated with the alternative set in (5a), containing the conjunctive alterna-tive, as well as the individual disjuncts (the so-called domain alternatives). Incalculating the result of applying the exhaustification operator, we first needto identify which of the alternatives are innocently excludable. As mentionedabove, an alternative is innocently excludable only if its negation can beadded to the assertion without resulting in a meaning that entails anotheralternative. Note that neither domain alternative satisfies this condition asthe exclusion of one results in a meaning that entails the other and vice versa.Since neither domain alternative is innocently excludable, the exhaustifica-tion proceeds with respect to a subset of the alternative set, namely the set

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containing the conjunctive alternative only. The result of applying Exh willproduce the enriched meaning in (5b):

(5) Mary invited John or Bill.

a. Alt(Mary invited John or Bill) = {Mary invited John and Bill, Maryinvited John, Mary invited Bill}

b. Exh[Mary invited John or Bill] = Mary invited John or Bill & Marydidn’t invite John and Bill

It is worth noting, however, that a sentence like (5) does not always have theenriched meaning in (5b); depending on the context, the implicature ‘Marydidn’t invite both John and Bill’ may or may not be present. Assuming thisgrammatical approach to scalar implicatures, there are a few ways to thinkabout the optionality of implicatures. One option is to take exhaustificationto be an obligatory operation across the board and appeal to a notion ofalternative pruning in order to derive non-enriched meanings. Under thisapproach the difference between the inclusive (non-enriched) and exclusive(enriched) use of disjunction would be the result of what alternative set Exhmakes reference to: for the inclusive reading the alternative set would beempty, whereas for the exclusive reading the alternative set would be as in(5a).3 Another option is to assume that the exhaustification operator is itselfoptional. Under this approach, a sentence like (5) can be said to be ambiguousbetween the two LFs in (6).

(6) Mary invited John or Bill.

a. Mary invited John or Bill inclusiveb. Exh[Mary invited John or Bill] exclusive

For the purposes of this paper I will couch the analysis in terms of a hybridapproach to exhaustification, wherein both alternative pruning and option-ality of exhaustification can be employed, although it is worth noting thatassuming exhaustification is optional is akin to assuming that all alternativesare pruned.

3 The prejacent of Exh is itself an alternative but for presentational purposes I will not includeit in the alternative set.

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4 Plain versus complex disjunctions

Most languages have more than one way of conveying disjunction. For ex-ample, in English we find or and either or, in French ou, ou ou and soit soit,in Romanian sau, ori, ori ori, fie fie, in Hungarian vagy, vagy vagy and akárakár and in German we encounter oder and entweder oder. One of the maindifferences between these ways of conveying disjunction within a languageboils down to whether the disjunction is interpreted inclusively or exclusivelyin positive contexts.4 The exclusive interpretation comes about whenever theuse of a disjunction gives rise to the inference that it cannot be the case thatboth disjuncts are true. For example, in English both or and either or cangive rise to the inference ‘not both’.

(7) a. Mary will visit John or Bill. ; Mary won’t visit both.b. Mary will visit either John or Bill. ; Mary won’t visit both.

The difference between these two disjunctions comes when we try to cancelthis inference. Whereas (8a) can be continued with ‘possibly both’, (8b) cannot,at least not as easily, which has been taken to suggest that either or is thenatural language counterpart of the logical exclusive disjunction, while or isthe natural language counterpart of the logical inclusive disjunction.5

(8) a. Mary will visit John or Bill. . . . possibly bothb. Mary will visit either John or Bill. . . . #possibly both

The same contrast is observed cross-linguistically. In French, for example,the difference between the disjunctions ou and soit soit is parallel to thedifference noted above for English: soit soit gives rise to the exclusivityinference more robustly than ou, given that a continuation which contradictsthe scalar inference is significantly less natural if the complex disjunctionsoit soit is used.

(9) a. Marie ira au cinéma lundi ou mardi.‘Marie will go to the movies on Monday or Tuesday.’

b. Absolument! Et elle ira même à la fois lundi ET mardi.‘Absolutely! She will even go both days.’

4 Nonetheless, in the languages that make a three and even a four-way distinction, it remainsto be understood what other levels of variation there are.

5 Nicolae & Sauerland (2016) provide experimental evidence in support of this claim.

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(10) a. Marie ira au cinéma soit lundi soit mardi.‘Marie will go to the movies either on Monday or Tuesday.’

b. #Absolument! Et elle ira même à la fois lundi ET mardi.‘Absolutely! She will even go both days.’

4.1 Spector’s (2014) account of complex disjunction PPIs

Spector (2014) argues that cross-linguistically, complex disjunctions alsoexhibit PPI-like behavior.6 For example, French soit soit cannot receive anarrow scope interpretation with respect to a c-commanding negation, (11),but it can if the negation is further embedded under a downward-entailingoperator, (12).

(11) Pierre ne parle pas soit allemand soit anglais.‘Pierre doesn’t speak soit German soit English.’

a. Pierre doesn’t speak German, or he doesn’t speak English.b. *Pierre doesn’t speak either German or English.

(12) Je n’emmène jamais Marie au cinéma sans qu’elle ait demandé lapermission soit à son père soit à sa mère.‘I never bring Marie to the movies without her having asked permissionfrom her father or from her mother.’

Spector claims that these two distributional restrictions observed with com-plex disjunctions, obligatory scalar implicatures and restriction to upwardentailing environments, should be seen as the result of the same underlyingmechanism. In particular, he argues that complex disjunctions should be an-alyzed as scalar elements that obligatorily trigger exhaustification. As alreadydiscussed, scalar implicatures are the result of applying the Exh operator, asrepeated below:

(13) a. Exh[p ∨ q] = (p ∨ q)∧¬(p ∧ q)b. Exh[Mary will visit John or Bill] = Mary will visit John or Bill &

Mary won’t visit John and Bill

Why is the scalar implicature ‘not both’ associated with soit soit and either orstronger than that of ou and or, respectively? The claim is that unlike plaindisjunction, which is ambiguous between the two LFs in (14), the complex

6 Except for English either or which is not a PPI, as pointed out by Spector himself. I will returnto a discussion of either or in the concluding remarks of the paper.

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disjunction is unambiguously interpreted with an Exh operator, that is, onlythe LF in (14b) is available.

(14) a. [p ∨ q] ✓ou, 7soit soitb. Exh[p ∨ q] ✓ou, ✓soit soit

This straightforwardly accounts for the obligatory presence of an implicaturewith complex but not plain disjunctions. In order to account for the PPIbehavior of complex disjunctions, Spector has to furthermore assume thatthe application of Exh is constrained by a pragmatic economy conditionwhich dictates that the contribution of Exh must give rise to strengthening(cf. Fox & Spector 2018, among others).

(15) An occurrence of Exh in a sentence S is not licensed if eliminatingthis occurrence leads to a sentence S’ which entails or is equivalentto S.

If soit soit is analyzed as a disjunction (with conjunction as its alterna-tive) which triggers obligatory exhaustification, the PPI-like behavior of thiselement comes out straightforwardly. Since disjunction is weaker than con-junction only in upward entailing environments, only in these environmentsdoes the contribution of Exh lead to strengthening, (16) versus (17), hence therestriction of soit soit to upward entailing environments.

(16) Exh[p ∨ q]a. Alt(p ∨ q) = {p,q,p ∧ q}b. Exh[p ∨ q] = (p ∨ q)∧¬(p ∧ q)

(17) Exh[¬(p ∨ q)]a. Alt(¬(p ∨ q)) = {¬p,¬q,¬(p ∧ q)}b. Exh[¬(p ∨ q)] = ¬(p ∨ q)

4.2 The problem of plain disjunction PPIs

To recapitulate, plain and complex disjunctions are distinguished by thestrength of the scalar implicature not both: in the case of plain disjunctionthe implicature is easier to cancel than in the case of complex disjunction.Spector (2014) argues that the strength of the implicature and the positivepolarity status of complex disjunctions have the same source, namely anobligatory association with an exhaustification operator whose insertion is

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subject to the economy condition in (15). In other words, being a PPI is tiedto the presence of a scalar implicature. While this analysis works well forcomplex disjunctions, it leaves wide open the question of how to analyzeplain disjunction PPIs like French ou, given that plain disjunctions are onlyoptionally associated with scalara implicatures. The goal going forward isto argue that, more generally, PPIs are elements which are lexically markedas triggering obligatory exhaustification. I choose to encode obligatory ex-haustification as a lexical requirement (similarly to the approach taken inChierchia 2013 to deal with NPIs). As it is generally the case with polarityphenomena, it is difficult to explain why certain elements exhibit polaritysensitivity in some languages but not in others, so for the purpose of thispaper I will simply have to stipulate that certain disjunctions lexically encodethe need for obligatory exhaustification whereas others do not.

The first step in the analysis will be to render the scalar implicatureoptional in the case of plain disjunctions. In this spirit, I follow previousauthors (Fox & Katzir 2011, Crnic, Chemla & Fox 2015) who have argued thatalternatives can be pruned, namely that exhaustification can proceed withrespect to a subset of the set of innocently excludable alternatives. I willadopt this assumption and propose that plain disjunction may prune thescalar alternative from its alternative set, whereas complex disjunction maynot.7

In other words, a plain disjunction can associate with either of the al-ternative sets in (18), whereas a complex disjunction is restricted to the fullalternative set in (18a). The argument for restricting complex disjunctionsto the set in (18a) comes from the fact that they appear to obligatorily giverise to a scalar implicature (cf. Spector 2014, Nicolae & Sauerland 2016). Fromhere on out I will annotate these different alternative sets asAltS andAltDrespectively. I will furthermore subscript the exhaustification operator with S(for scalar) or D (for domain) to indicate which alternative set the operatorassociates with.

(18) a. AltS(p ∨ q) = {p,q,p ∧ q}b. AltD(p ∨ q) = {p,q}

7 As already discussed in Fox & Katzir 2011 and Ivlieva 2012, pruning of alternatives mustbe constrained. For example, pruning the conjunctive alternative should not be allowed ifrecursive exhaustification is employed, for the end result would be a conjunctive meaning. Iwill adopt a constraint on pruning which requires the result of exhaustification with respectto a pruned set of alternatives to give rise to a meaning that could not have been expressedby a (stronger) alternative obtained via lexical replacement.

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The claim will be that plain disjunction PPIs (French ou), like complexdisjunction PPIs (French soit soit), and unlike plain disjunction non-PPIs (En-glish or) trigger obligatory exhaustification, an operation which is governedby the economy condition ruling out vacuous instances of Exh (cf. (15)). Thedifference between the two types of disjunction PPIs will be that plain dis-junction PPIs trigger exhaustification with respect to a (possibly) differentset of alternatives, namely the one in (18b). Let’s consider the predictions ofsuch an account. Under negation, the domain alternatives will be as in (19a).Since the alternatives are entailed by the assertion, the exhaustification willbe vacuous, as in (19b).8

(19) ExhD[¬(p ∨ q)]a. AltD(¬(p ∨ q)) = {¬p,¬q}b. ExhD[¬(p ∨ q)] = ¬(p ∨ q)

So far this is a welcome result. The economy condition on Exh rules it outwhenever its contribution does not lead to strengthening. Since ou triggersobligatory exhaustification, the vacuity of Exh under negation delivers theunacceptability of ou under negation. On the other hand, since English ordoesn’t obligatorily trigger exhaustification, no problem will arise in a DEenvironment since an LF without Exh is acceptable.

Unfortunately, a problem arises once we turn to UE contexts. Recall thatwe are assuming exhaustification takes place only with respect to innocentlyexcludable alternatives. Since the alternatives in (20a) are not innocentlyexcludable, the exhaustification of the assertion with respect to this set willbe vacuous, as in (20b):

(20) ExhD[p ∨ q]a. AltD(p ∨ q) = {p,q}b. ExhD[p ∨ q] = p ∨ q

What this account predicts then is that ou should also be ruled out in UEcases given that here too the result of exhaustification is vacuous. This isobviously a wrong prediction and a solution needs to be found. A possibleway of avoiding the vacuity in (20b) would be by exhaustifying with respecttoAltS , as in (21):

8 Another option of course would be to exhaustify below the negation, ¬[ExhD[p ∨ q]].The domain alternatives in this configuration are not innocently excludable, rendering theexhaustification vacuous.

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(21) ExhS[p ∨ q]a. AltS(p ∨ q) = {p,q,p ∧ q}b. ExhS[p ∨ q] = (p ∨ q)∧¬(p ∧ q)

The issue with such an approach is that it would mean that in UE contextsthe plain disjunction would always give rise to a scalar implicature, giventhe necessity of including the scalar alternative in order to avoid vacuousexhaustification. That is clearly not the case given the facts outlined ear-lier when describing the difference between plain and complex disjunction,namely that ou does not obligatorily trigger a scalar implicature. Employingthis mechanism would essentially obfuscate the difference between plain andcomplex disjunction, making a wrong empirical prediction.

The problem of plain disjunction PPIs thus remains. With the tools avail-able thus far, we have no way of deriving the PPI behavior of plain disjunctionssimply by appealing to obligatory exhaustification.

5 A solution to the problem of plain disjunction PPIs

We saw in the previous section that the ban on vacuous exhaustificationmakes the wrong prediction when it comes to the distribution of plaindisjunction PPIs in UE contexts. In this section I will argue that this problemwill not arise if we adopt the proposal in Meyer 2013 which takes uncertaintyimplicatures, such as the one in (22), normally thought of as arising viapragmatic principles (e.g., via Grice’s Cooperative Principle), to also be derivedin the grammar, similarly to scalar implicatures.

(22) Mary invited John or Paul. ; But I don’t know which.

Meyer’s claim is that assertively used sentences contain a covert doxasticoperator which is adjoined at the matrix level at LF (cf. also Kratzer & Shi-moyama 2002, Chierchia 2006 and Alonso-Ovalle & Menéndez-Benito 2010for similarly minded proposals). She calls this operator K (following Gaz-dar (1979)) and gives it the semantics in (23). I represent this operator as anecessity modal throughout the remainder of the text.

(23) �2xp� = λw.∀w′ ∈ Dox(x)(w) : p(w′)w′ ∈ Dox(x)(w) iff given the beliefs of x in w, w’ could be the actualworld.

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By bringing this operator into the grammar we can derive the uncertainty im-plicatures of disjunction similarly to how we derive its scalar implicature, viathe application of an exhaustification operator. This implicature is obtainedby having the exhaustifier take scope over the doxastic operator, as in (24):

(24) ExhD[2[p ∨ q]]a. AltD(2[p ∨ q]) = {2p,2q}b. ExhD[2[p ∨ q]] = 2(p ∨ q)∧¬2p ∧¬2q

Exhaustifying with respect to this set of alternatives will deliver uncertaintyimplicatures about the two domain alternatives, given that the domain alter-natives are now innocently excludable. For a sentence such as ‘Mary visitedJohn or Bill’, the enriched meaning in (24b) will amount to ‘I am certain thatMary visited one of the two, but it’s possible she didn’t visit John and it’spossible she didn’t visit Bill’, hence the uncertainty with respect to the statusof the individual disjuncts.

Adopting this way of deriving uncertainty implicatures allows for a uni-form approach to implicatures, both scalar and uncertainty. Most importantlyfor our purposes, however, it straightforwardly derives the acceptability ofelements triggering obligatory (and thus strengthening) exhaustification inUE cases. Notice that the enriched meaning in (24b) is stronger than the non-enriched meaning. In other words, the exhaustification is no longer vacuous,rendering the PPI disjunction ou acceptable in UE contexts.

Before concluding this section it is worth checking that under negation thepresence of the doxastic operator does not lead to any (undesired) strength-ening. The result of exhaustification in (25) will be vacuous given that thesister of ExhD, which is equivalent to 2¬p ∧2¬q, by epistemic logic, entailseach of the alternatives.

(25) ExhD[2¬[p ∨ q]]a. AltD(2¬[p ∨ q]) = {2¬p,2¬q}b. ExhD[2¬[p ∨ q]] = 2¬(p ∨ q)

We have successfully shown why plain disjunction PPIs are unacceptableunder negation but acceptable in UE environments if we analyze them aselements that trigger obligatory exhaustification and allow pruning of thescalar alternative.

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5.1 Re-assessing complex disjunction

Taking uncertainty implicatures to be derived in the semantics, as Meyer(2013) suggests, requires us to re-evaluate the case of complex disjunction.Whereas plain and complex disjunctions differ with respect to the availabilityof their scalar implicature, they do not exhibit the same variability when weturn to the uncertainty implicature. Both types of disjunction give rise to thisimplicature just as robustly:

(26) Mary visited (either) John or Bill. ; But I don’t know which.

Since the source of the uncertainty implicature is the interaction betweenthe doxastic operator and the domain alternatives, we must check whathappens when exhaustification occurs with respect to both scalar and domainalternatives in the presence of the doxastic operator. In particular, we need toensure that the scalar implicature is still derived. Since complex disjunctiondoes not allow its scalar alternative to be pruned, we need to check the resultof exhaustification via ExhS , as in (27):

(27) ExhS[2[p ∨ q]]a. AltS(2[p ∨ q]) = {2p,2q,2[p ∧ q]}b. ExhS[2[p ∨ q]] = 2(p ∨ q)∧¬2p ∧¬2q ∧¬2(p ∧ q)

Observe that while the uncertainty implicature is still derived, the scalarimplicature is weakened (last conjunct), deriving instead something muchweaker, namely that it’s possible not both. This is not ideal since the differ-ence between plain and complex disjunction is no longer derived.

There is, however, another possible LF, one where the exhaustificationis embedded, as in (28). Appealing to embedded exhaustification allows usto derive the stronger scalar implicature, namely that it is necessarily notthe case that both are true. It’s crucial of course that we also employ matrixexhautification or else the uncertainty implicature would not be derived.

(28) ExhS[2[ExhS[p ∨ q]]]a. AltS(p ∨ q) = {p,q,p ∧ q}b. ExhS[p ∨ q] = (p ∨ q)∧¬(p ∧ q)c. AltS(2ExhS[p ∨ q]) = {2ExhS p,2ExhS q,2ExhS[p ∧ q]}

= {2[p ∧¬q],2[q ∧¬p],2[p ∧ q]}d. ExhS2ExhS[p ∨ q]

= 2[p ∨ q]∧2¬[p ∧ q]∧¬2[p ∧¬q]∧¬2[q ∧¬p]

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If we think of (27) and (28) as two competing LFs, we can invoke the StrongestMeaning Hypothesis (Dalrymple et al. 1998) and end up with (28) as thepreferred reading, as desired. This might turn out to be good for anotherreason: complex disjunction can, sometimes, lack the exclusivity inference,and instead of attributing this behavior to the optionality of scalar exhaustifi-cation (which would go against the proposal in Spector 2014), we can insteadclaim that the LF in (27) is employed.9

This section has shown that adopting an account of uncertainty implica-tures that takes them to be derived in the grammar allows us to make thesame predictions as before, namely that complex disjunctions give rise to ascalar implicature.

5.2 PPIs without uncertainty implicatures?

A prediction made by this account is that non-PPI disjunctions, which donot trigger obligatory exhaustification, should allow for continuations thatcontradict the uncertainty inference. This prediction appears to be correct,given the felicity of the discourse below.

(29) Mary talked with John or Paul. In fact, she talked with both.

If I’m certain that Mary talked with both John and Paul, then it can’t bepossible that she didn’t talk with John, nor can it be possible that shedidn’t talk with Paul, i.e., the uncertainty inference must be false. We sawabove that exhaustification with respect to the domain alternatives will giverise to a strengthened meaning only in the presence of a speaker-orienteddoxastic operator. By relying on the presence of this operator to derive astrengthened meaning upon exhaustification, we make the prediction thata plain disjunction PPI will only ever be able to receive an interpretation inan UE context if it gives rise to an uncertainty inference. This is a wrong

9 Note that the LF in (27) is also a possible interpretation for plain disjunctions. One could infact conceive of framing the difference between plain and complex disjunctions in termsof matrix versus embedded exhaustification, i.e., (27) versus (28), respectively. If this ap-proach were adopted, however, more would have to be said about the need for embeddedexhaustification with complex disjunction. One avenue to pursue would be to argue thatthe disjuncts in complex disjunction are interpreted exhaustively, support for which we candraw from the fact that such disjunctions are usually associated with prosodic focus on thedisjuncts. The LF for complex disjunctions would thus be ExhS[2[ExhDp ∨ExhDq]] ratherthan (28), which gives rise to the same meaning as in (28d).

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prediction given that the French equivalent of (29), given in (30), is perfectlyfelicitous.

(30) Marie a parlé à Jean ou Paul. En fait, elle a parlé aux deux.

The continuation ‘in fact both’, 2(p∧q), is incompatible with the uncertaintyimplicature obtained by the application of the exhaustification operator,provided in (31). So what gets us into trouble with the continuation is preciselywhat allowed a PPI disjunction to receive an interpretation in UE cases (underthe analysis pursued here), namely the strengthening via the uncertaintyimplicature.

(31) ¬2p ∧¬2q

What we need then is to derive a strengthened meaning of the disjunctionthat will be compatible with the continuation in (29). I argue that invokingboth embedded and matrix exhaustification with respect to the domainalternatives will yield a meaning compatible with a situation in which bothare true.

(32) ExhD[2[ExhD[p ∨ q]]]a. AltD(p ∨ q) = {p,q}b. ExhD[p ∨ q] = p ∨ qc. AltD(2[ExhD[p ∨ q]]) = {2ExhD p,2ExhD q}

= {2(p ∧¬q),2(q ∧¬p)}d. ExhD[2[ExhD[p ∨ q]]] = 2(p ∨ q)∧¬2(p ∧¬q)∧¬2(q ∧¬p)

≡ 3p ∧3q

The equivalence in (32d) holds because if it’s true that Mary necessarily talkedwith John or Paul but that she didn’t necessarily talk only with John, thenit follows that it’s possible that she talked with Paul, and vice versa. Thisrecursively enriched meaning is now compatible with a situation in whichboth p and q must be true.10

In summary, we can now understand how it is possible for an unembed-ded plain disjunction that exhibits PPI behavior to lack both a scalar and

10 This same approach is independently adopted by Crnic, Chemla & Fox (2015) to account forthe observation that sentences with disjunction in the scope of a universal quantifier, EveryA is P or Q, tend to give rise to distributive inferences that each of the disjuncts holds ofat least one individual in the domain of the quantifier, Some A is P & Some A is Q in theabsence of plain negated inferences, Not every A is P & Not every A is Q .

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an uncertainty implicature, and yet still count as strengthened for the pur-poses of satisfying the economy condition on non-vacuous exhaustification.Whereas in the case of English or we might have dealt with the acceptability ofthis continuation by simply stating that the exhaustification is not obligatory,in the case of the French ou, a PPI, such an approach is not possible since thedisjunction triggers obligatory exhaustification. Suspending exhaustificationin this case should not be an option for then we would expect exhaustifica-tion to also be suspendable under negation, therefore no longer deriving theunacceptability of disjunction in such contexts.

5.3 Overgeneration issues

The analysis as presented thus far predicts that ou should be unacceptableunder any downward entailing operator. This is a wrong prediction, as evi-denced by the data in (33) where ou can receive a narrow scope interpretationwith respect to DE operators such as ‘few’, ‘less than ten’, as well as in theantecedent of conditionals and the restrictor of universals.

(33) a. Peu de/Moins de dix étudiants parlent espagnol ou italien.‘Few/Less than ten students speak Spanish or Italian.’

b. Si Marie a pris un cours de maths ou de physique ce semestre,elle réussira l’examen.‘If Mary took math or physics this semester, she’ll pass the exam.’

c. Tout étudiant qui a pris un cours de maths ou de physique réus-sira l’examen.‘Every student who took math or physics passed the exam.’

This is reminiscent of the behavior of strong NPIs like English ‘until’ and ‘inweeks’ which, unlike weak NPIs such as ‘any’ and ‘ever’, are acceptable undernegation but not in the environments above (Homer 2009, Gajewski 2011,Chierchia 2013).

(34) a. No students have attended this course in weeks.b. *Few/less than ten students have attended this course in weeks.c. *If Mary has attended this course in weeks, she should inform us.d. *Every student who has attended this course in weeks will pass.

The claim put forth by Gajewski (2011) to account for this contrast is thatstrong, but not weak, NPIs see not only the truth-conditional meaning but

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also the non-truth-conditional meaning of their licensors; this includes what-ever implicatures and/or presuppositions the licensor may give rise to. Forexample, ‘few’ in (34b) gives rise to the implicature that some students havecompleted their homework. While ‘few’ on its own creates a DE environment,when conjoined with its implicature, the overall monotonicity is no longerdownward-entailing, but rather non-monotonic. Since the mechanism of NPIlicensing requires them to occur in a DE environment, the unacceptability ofstrong NPIs under operators like ‘few’ falls out immediately (see Chierchia2013 for the details of the account). The same can be argued for ‘less thann’, while in the case of the antecedent of conditionals and the restrictor ofuniversals, what disrupts the downward-entailing monotonicity is the pre-supposition associated with the conditional and the quantifier, respectively.

I will argue that the acceptability of ou in (33) can be accounted for viathe same logic. In particular, I will propose that PPIs like ou look at boththe truth-conditional and non-truth-conditional meaning of their licensor,making them the counterparts of strong NPIs within the PPI domain.11 In theremainder of the section I will demonstrate how this works by focusing onthe felicity of PPI disjunctions in the restrictor of universal quantifiers.

To reiterate, the disjunction in (33c) is interpreted in the restrictor of theuniversal. If we only looked at the truth-conditional meaning when calculatingthe result of applying Exh at the matrix level in (33c) we would expect only awide-scope interpretation of ou, given that the restrictor of universals createsa DE environment. In order to account for the acceptability of PPIs in therestrictor of the universal, I will make the aforementioned assumption thatpresuppositions can enter into the calculation of exhaustification. The crucialobservation is that the restrictor of a universal quantifier is Strawson-DE(cf. von Fintel 1999), due to the fact that universal quantifiers contribute apresupposition of existence, provided in (35).

(35) Every student who took math or physics passed the exam.defined if: Some student(s) took math or physics.

I will argue that the exhaustification occurs with respect to the conjunctionof the assertion and the presupposition, as in (36). Given the alternatives in

11 This approach is supported by an observation made by van der Wouden (1997) that cross-linguistically, PPIs exhibit the same variation familiar from the domain of NPIs. For example,just like ou is the PPI counterpart of strong NPIs, the Dutch allerminst ‘not in the least’ is aPPI that is unacceptable under operators like ‘few’ and ‘less than n’, namely the counterpartof weak NPIs like ‘any’ and ‘ever’.

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(36a), the result of exhaustification is the enriched meaning in (36b), withthe inference that it’s possible that nobody took math and it’s possible thatnobody took physics.

(36) ExhD[2∀x[[p ∨ q](x)→ r(x)]∧2∃x[[p ∨ q](x)]]a. AltD(2∀x[[p ∨ q](x)→ r(x)]∧2∃x[[p ∨ q](x)])

=

2∀x[p(x)→ r(x)]∧2∃x[p(x)],2∀x[q(x)→ r(x)]∧2∃x[q(x)]

b. ExhD[2∀x[[p ∨ q](x)→ r(x)]∧2∃x[[p ∨ q](x)]]

= 2∀x[[p ∨ q](x)→ r(x)]∧2∃x[[p ∨ q](x)]∧¬2∃x[p(x)]∧¬2∃x[q(x)]

= 2∀x[[p ∨ q](x)→ r(x)]∧2∃x[[p ∨ q](x)]∧3¬∃x[p(x)]∧3¬∃x[q(x)]

What (36) shows is that exhaustification of the domain alternatives is non-vacuous as soon as the presupposition is taken into account, meaning thatou is correctly predicted to survive in the restrictor of universals. A similarargument can be made for the antecedent of conditionals which carry thepresupposition that the restrictor is a possibility. As for quantifiers like‘few’ and ‘less than n’, the implicatures these operators give rise to play thesame role as the presupposition of ‘every’ did above, namely they create anon-monotonic environment whereby the exhaustification triggered by thePPI is non-vacuous.

5.4 Rescuing by a second negation

Recall that (37) is unambiguously interpreted with disjunction taking widescope over negation.

(37) Marie n’a pas pris un cours de maths ou de physique ce semestre.‘Mary either didn’t take math or she didn’t take physics this semester.’

The observation is that if we further embed (37) in a DE context as in (38b),the disjunction can be interpreted in the scope of negation. For example,beyond the wide scope reading of disjunction, (38a) also has the possiblereading that the students who took neither math nor physics failed the exam.

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Deriving the positive polarity behavior of plain disjunction

(38) a. Tout étudiant qui n’a pas pris de cours de maths ou de physiquea raté l’examen.‘Every student who took neither math onr physics failed theexam.’

b. Si Marie n’a pas pris un cours de maths ou de physique, elle araté l’examen.‘If Mary took neither math nor physics, she failed the exam.’

Being embedded under two DE operators is equivalent to being in apositive environment for the purposes of exhaustification: (i) the alternativesare stronger than the assertion, and (ii) the result of exhaustification is notvacuous. Below I illustrate this for the case of a negated disjunction in therestrictor of a universal:

(39) ExhD[2∀x[¬[p ∨ q](x)→ r(x)]∧2∃x[¬[p ∨ q](x)]]a. AltD(2∀x[¬[p ∨ q](x)→ r(x)]∧2∃x[¬[p ∨ q](x)])

=

2∀x[¬p(x)→ r(x)]∧2∃x[¬p(x)],2∀x[¬q(x)→ r(x)]∧2∃x[¬q(x)]

b. ExhD[2∀x[¬[p ∨ q](x)→ r(x)]∧2∃x[¬[p ∨ q](x)]]

= 2∀x[¬[p ∨ q](x)→ r(x)]∧2∃x[¬[p ∨ q](x)]∧¬2∀x[¬p(x)→ r(x)]∧¬2∀x[¬q(x)→ r(x)]

The result of exhaustification is the inference that it’s not necessarily thecase that every student who didn’t take math failed the exam, and similarly,that it’s not necessarily the case that every student who didn’t take physicsfailed the exam.

6 Overview and outlook

In this paper I argued that the PPI behavior of plain disjunction should beanalyzed as an interplay between a semantic requirement for obligatoryexhaustification and an economy condition which prevents vacuous exhaus-tification, building on the analysis provided by Spector (2014) to account forthe PPI behavior of complex disjunctions cross-linguistically. I showed thatonce this system is adopted, coupled with a condition on alternative pruningand the claim that exhaustification can take scope over a covert doxasticoperator, we can straightforwardly derive the restricted distribution of plaindisjunction PPIs. Specifically, I argued that plain, but not complex, disjunction

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allows the pruning of its conjunctive alternative, using as evidence the con-trast between these two types of disjunction when it comes to the optionalityof their scalar implicature. This analysis was shown to derive the inability ofPPIs to scope under a local negation, their ability to be rescued by a secondDE operator, as well as the fact that plain disjunction PPIs can survive in theabsence of uncertainty implicatures.

What was not discussed in this paper was the fact that ou can take narrowscope with respect to an extra-clausal negation, as shown in (40).

(40) Paul ne pense pas que Marie ait invité Pierre ou Julie à dîner.‘Paul doesn’t think that Marie invited Pierre or Julie for dinner.’

a. Paul doesn’t think that Marie invited Pierre or he doesn’t thinkthat Marie invited Julie to dinner. or>not

b. Paul doesn’t think that Marie invited Pierre and he doesn’t thinkthat Marie invited Julie to dinner. not>or

The analysis, as laid out in this paper, predicts ou to be unacceptable undernegation, regardless of its locality. There are, as I see, a couple of ways totackle this problem. One possibility would be to invoke two levels of recursiveexhaustification, namely below and above the negation; doing so woulddeliver the narrow scope reading of the disjunction, while also employingexhaustification (cf. Nicolae 2016 for the details of such an analysis). The factthat we are dealing with a clause boundary between the negation and thedisjunction might be the clue to understanding why the application of theexhaustification operator can be said to be non-vacuous when the negation isextra-clausal, but not when it is local to the disjunction. There are, however,issues with this approach, pertaining to alternative selection as well as thestrengthening condition on exhaustification, and space limitations preventme from discussing this further. Yet another possibility for dealing with thiscontrast would be to look at how the embedding predicate interacts with thec-commanding negation and whether this interaction leads to any inferencesthat may satisfy the non-vacuity condition on the exhaustifying operator. Ihope to tackle these possibilities on another occasion.

Any discussion of positive polarity would not be complete without atleast a mention of how it can be integrated within the larger polarity sys-tem. Presently, the proposal offered here to account for the positive polaritybehavior of disjunction is not immediately compatible with the proposaloffered in Chierchia 2013 to account for the distribution of NPIs. The com-

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Deriving the positive polarity behavior of plain disjunction

parison is done with Chierchia’s account for the simple reason that this isthe most recent large scale discussion of the polarity system that is alsocouched within the grammatical approach to implicature calculation. Thereason for the incompatibility is due to the fact that Chierchia does notemploy a contradiction-free method of exhaustification, namely one that onlytakes into account innocently excludable alternatives. In fact, his system forderiving the unacceptability of NPIs in non-DE environments hinges on thecontradiction incurred by negating non-innocently excludable alternatives.On the other hand, the analysis employed here to account for the distributionof PPIs hinges on the fact that exhaustification is contradiction-free, for elseelements that are PPIs would systematically be predicted to be unavailablein both DE and UE contexts. While the current paper is not the appropriatevenue for a deep investigation of these issues, I would like to leave the readerwith some semblance of a resolution to this conflict by pointing out thatanother implicature-based account of NPIs that is also compatible with thepresent analysis of PPIs can be found in the work of Crnic (2014). Crnicanalyzes NPIs as end-of-scale indefinites that obligatorily trigger exhausti-fication via an even-like operator (as opposed to the Chierchia’s approachwhich takes NPIs to trigger obligatory exhaustification via Exh). Connected tothis point, it is also worth mentioning the behavior of either or in English,a complex disjunction that does not exhibit PPI behavior (see fn. 6). This isprima facie a problem since as a complex disjunction, either or must triggerobligatory exhaustification, which in turn should result in unacceptabilityunder negation. What seems to be at play is the fact that either can alsofunction as an NPI, with a meaning not much different from that of any:

(41) Mary didn’t invite either/any of them.

One possible avenue for future research would be to argue that either oris a polarity sensitive item, but that unlike run of the mill NPIs and PPIs,it does not discriminate between the type of exhaustifier it can associatewith; while it obligatorily triggers exhaustification, the exhaustifier can beeither Exh, delivering its behavior in UE contexts, or Crnic’s NPI exhaustifier,delivering its acceptability in DE environments. In other words, a sentencewith either or is ambiguous between an LF with Exh and an LF with even,the choice being determined by whichever one leads to a non-contradictorymeaning in the end.

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Andreea C. Nicolae

Leibniz-Zentrum Allgemeine

Sprachwissenschaft

Schützenstraße 18

D–10117 Berlin, Germany

nicolae@leibniz-zas.de

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